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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > reprle | Structured version Visualization version GIF version |
Description: Upper bound to the terms in the representations of 𝑀 as the sum of 𝑆 nonnegative integers from set 𝐴. (Contributed by Thierry Arnoux, 27-Dec-2021.) |
Ref | Expression |
---|---|
reprval.a | ⊢ (𝜑 → 𝐴 ⊆ ℕ) |
reprval.m | ⊢ (𝜑 → 𝑀 ∈ ℤ) |
reprval.s | ⊢ (𝜑 → 𝑆 ∈ ℕ0) |
reprf.c | ⊢ (𝜑 → 𝐶 ∈ (𝐴(repr‘𝑆)𝑀)) |
reprle.x | ⊢ (𝜑 → 𝑋 ∈ (0..^𝑆)) |
Ref | Expression |
---|---|
reprle | ⊢ (𝜑 → (𝐶‘𝑋) ≤ 𝑀) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fveq2 6409 | . 2 ⊢ (𝑎 = 𝑋 → (𝐶‘𝑎) = (𝐶‘𝑋)) | |
2 | fzofi 13024 | . . 3 ⊢ (0..^𝑆) ∈ Fin | |
3 | 2 | a1i 11 | . 2 ⊢ (𝜑 → (0..^𝑆) ∈ Fin) |
4 | reprval.a | . . 3 ⊢ (𝜑 → 𝐴 ⊆ ℕ) | |
5 | reprval.m | . . 3 ⊢ (𝜑 → 𝑀 ∈ ℤ) | |
6 | reprval.s | . . 3 ⊢ (𝜑 → 𝑆 ∈ ℕ0) | |
7 | reprf.c | . . 3 ⊢ (𝜑 → 𝐶 ∈ (𝐴(repr‘𝑆)𝑀)) | |
8 | 4, 5, 6, 7 | reprsum 31203 | . 2 ⊢ (𝜑 → Σ𝑎 ∈ (0..^𝑆)(𝐶‘𝑎) = 𝑀) |
9 | 4 | adantr 473 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ (0..^𝑆)) → 𝐴 ⊆ ℕ) |
10 | 4, 5, 6, 7 | reprf 31202 | . . . . 5 ⊢ (𝜑 → 𝐶:(0..^𝑆)⟶𝐴) |
11 | 10 | ffvelrnda 6583 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ (0..^𝑆)) → (𝐶‘𝑎) ∈ 𝐴) |
12 | 9, 11 | sseldd 3797 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ (0..^𝑆)) → (𝐶‘𝑎) ∈ ℕ) |
13 | 12 | nnrpd 12111 | . 2 ⊢ ((𝜑 ∧ 𝑎 ∈ (0..^𝑆)) → (𝐶‘𝑎) ∈ ℝ+) |
14 | reprle.x | . 2 ⊢ (𝜑 → 𝑋 ∈ (0..^𝑆)) | |
15 | 1, 3, 8, 13, 14 | fsumub 30084 | 1 ⊢ (𝜑 → (𝐶‘𝑋) ≤ 𝑀) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 385 ∈ wcel 2157 ⊆ wss 3767 class class class wbr 4841 ‘cfv 6099 (class class class)co 6876 Fincfn 8193 0cc0 10222 ≤ cle 10362 ℕcn 11310 ℕ0cn0 11576 ℤcz 11662 ..^cfzo 12716 reprcrepr 31198 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2354 ax-ext 2775 ax-rep 4962 ax-sep 4973 ax-nul 4981 ax-pow 5033 ax-pr 5095 ax-un 7181 ax-inf2 8786 ax-cnex 10278 ax-resscn 10279 ax-1cn 10280 ax-icn 10281 ax-addcl 10282 ax-addrcl 10283 ax-mulcl 10284 ax-mulrcl 10285 ax-mulcom 10286 ax-addass 10287 ax-mulass 10288 ax-distr 10289 ax-i2m1 10290 ax-1ne0 10291 ax-1rid 10292 ax-rnegex 10293 ax-rrecex 10294 ax-cnre 10295 ax-pre-lttri 10296 ax-pre-lttrn 10297 ax-pre-ltadd 10298 ax-pre-mulgt0 10299 ax-pre-sup 10300 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3or 1109 df-3an 1110 df-tru 1657 df-fal 1667 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2590 df-eu 2607 df-clab 2784 df-cleq 2790 df-clel 2793 df-nfc 2928 df-ne 2970 df-nel 3073 df-ral 3092 df-rex 3093 df-reu 3094 df-rmo 3095 df-rab 3096 df-v 3385 df-sbc 3632 df-csb 3727 df-dif 3770 df-un 3772 df-in 3774 df-ss 3781 df-pss 3783 df-nul 4114 df-if 4276 df-pw 4349 df-sn 4367 df-pr 4369 df-tp 4371 df-op 4373 df-uni 4627 df-int 4666 df-iun 4710 df-br 4842 df-opab 4904 df-mpt 4921 df-tr 4944 df-id 5218 df-eprel 5223 df-po 5231 df-so 5232 df-fr 5269 df-se 5270 df-we 5271 df-xp 5316 df-rel 5317 df-cnv 5318 df-co 5319 df-dm 5320 df-rn 5321 df-res 5322 df-ima 5323 df-pred 5896 df-ord 5942 df-on 5943 df-lim 5944 df-suc 5945 df-iota 6062 df-fun 6101 df-fn 6102 df-f 6103 df-f1 6104 df-fo 6105 df-f1o 6106 df-fv 6107 df-isom 6108 df-riota 6837 df-ov 6879 df-oprab 6880 df-mpt2 6881 df-om 7298 df-1st 7399 df-2nd 7400 df-wrecs 7643 df-recs 7705 df-rdg 7743 df-1o 7797 df-oadd 7801 df-er 7980 df-map 8095 df-en 8194 df-dom 8195 df-sdom 8196 df-fin 8197 df-sup 8588 df-oi 8655 df-card 9049 df-pnf 10363 df-mnf 10364 df-xr 10365 df-ltxr 10366 df-le 10367 df-sub 10556 df-neg 10557 df-div 10975 df-nn 11311 df-2 11372 df-3 11373 df-n0 11577 df-z 11663 df-uz 11927 df-rp 12071 df-ico 12426 df-fz 12577 df-fzo 12717 df-seq 13052 df-exp 13111 df-hash 13367 df-cj 14177 df-re 14178 df-im 14179 df-sqrt 14313 df-abs 14314 df-clim 14557 df-sum 14755 df-repr 31199 |
This theorem is referenced by: hgt750lemb 31246 |
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